Abelian variety isogeny class 2.73.p_gq over $\F_{73}$

• Label: 2.73.p_gq
• Base field: $\F_{73}$
• Dimension: $2$
• $p$-rank: $2$
• Ordinary: yes
• Supersingular: no
• Simple: no
• Geometrically simple: no
• Primitive: yes
• Principally polarizable: yes
• Contains a Jacobian: yes

Invariants

Base field:	$\F_{73}$
Dimension:	$2$
L-polynomial:	$( 1 + 2 x + 73 x^{2} )( 1 + 13 x + 73 x^{2} )$
	$1 + 15 x + 172 x^{2} + 1095 x^{3} + 5329 x^{4}$
Frobenius angles:	$\pm0.537340940774$, $\pm0.775177233176$
Angle rank:	$2$ (numerical)
Jacobians:	$90$

This isogeny class is not simple, primitive, ordinary, and not supersingular. It is principally polarizable and contains a Jacobian.

Newton polygon

This isogeny class is ordinary.

$p$-rank:	$2$
Slopes:	$[0, 0, 1, 1]$

Point counts

Point counts of the abelian variety

$r$	$1$	$2$	$3$	$4$	$5$
$A(\F_{q^r})$	$6612$	$29039904$	$150915144384$	$806477744509056$	$4297555707957824532$

Point counts of the curve

$r$	$1$	$2$	$3$	$4$	$5$	$6$	$7$	$8$	$9$	$10$
$C(\F_{q^r})$	$89$	$5449$	$387938$	$28398865$	$2073037769$	$151335174958$	$11047395180113$	$806460012526369$	$58871587614550994$	$4297625828361294889$

Jacobians and polarizations

This isogeny class is principally polarizable and contains the Jacobians of 90 curves (of which all are hyperelliptic):

• $y^2=13 x^5+10 x^4+59 x^3+15 x^2+63 x+35$
• $y^2=19 x^6+39 x^5+41 x^4+26 x^3+35 x^2+45$
• $y^2=x^6+51 x^4+50 x^3+3 x^2+55 x+54$
• $y^2=29 x^6+17 x^5+61 x^4+62 x^3+56 x^2+47 x+21$
• $y^2=25 x^6+34 x^5+25 x^4+62 x^3+66 x^2+39 x+9$
• $y^2=38 x^6+44 x^5+55 x^4+6 x^3+59 x^2+3 x+51$
• $y^2=5 x^6+35 x^5+64 x^4+28 x^3+28 x^2+29 x+21$
• $y^2=61 x^6+51 x^5+60 x^4+46 x^3+71 x^2+16 x+22$
• $y^2=52 x^6+7 x^5+42 x^4+67 x^3+21 x^2+22 x+39$
• $y^2=55 x^6+67 x^5+55 x^4+70 x^3+71 x^2+57 x+13$
• $y^2=55 x^6+53 x^5+64 x^4+13 x^3+67 x^2+66 x+46$
• $y^2=31 x^6+53 x^5+60 x^4+31 x^3+4 x^2+29 x+55$
• $y^2=25 x^6+22 x^5+62 x^4+26 x^3+28 x^2+43 x+32$
• $y^2=67 x^6+32 x^5+68 x^4+16 x^3+31 x^2+10 x+60$
• $y^2=41 x^6+58 x^5+18 x^4+18 x^3+54 x^2+23 x+25$
• $y^2=68 x^6+59 x^5+68 x^4+28 x^3+42 x^2+9 x+18$
• $y^2=57 x^6+31 x^5+38 x^4+15 x^3+68 x^2+44 x+50$
• $y^2=7 x^6+72 x^5+41 x^4+35 x^3+8 x^2+26 x+8$
• $y^2=65 x^6+57 x^5+12 x^4+34 x+28$
• $y^2=20 x^6+30 x^5+11 x^4+34 x^3+39 x^2+3$
• and

• $y^2=35 x^6+6 x^5+67 x^4+48 x^3+62 x^2+42 x+22$
• $y^2=43 x^6+47 x^5+28 x^4+60 x^3+15 x^2+34 x+69$
• $y^2=70 x^6+55 x^5+53 x^4+45 x^3+48 x^2+62 x+38$
• $y^2=29 x^6+3 x^5+64 x^4+57 x^3+44 x^2+43 x+11$
• $y^2=45 x^6+69 x^5+48 x^4+38 x^3+37 x^2+71 x+35$
• $y^2=50 x^6+4 x^5+67 x^4+31 x^3+37 x^2+6 x+9$
• $y^2=26 x^6+24 x^5+52 x^4+17 x^3+46 x^2+65 x+9$
• $y^2=56 x^6+24 x^5+x^4+60 x^3+68 x^2+51 x+64$
• $y^2=59 x^6+48 x^5+60 x^4+32 x^3+69 x^2+23 x+67$
• $y^2=4 x^6+47 x^5+40 x^4+27 x^3+55 x^2+38 x+4$
• $y^2=14 x^6+25 x^5+45 x^4+67 x^3+45 x^2+60 x+15$
• $y^2=72 x^6+37 x^5+44 x^4+20 x^3+13 x^2+22 x+11$
• $y^2=5 x^6+4 x^5+37 x^4+60 x^3+7 x^2+54 x+64$
• $y^2=39 x^6+35 x^5+62 x^4+63 x^3+48 x^2+64 x+3$
• $y^2=52 x^6+37 x^5+17 x^4+67 x^3+57 x^2+39 x+30$
• $y^2=16 x^6+12 x^5+44 x^4+69 x^3+13 x^2+48 x+44$
• $y^2=25 x^6+50 x^5+55 x^4+22 x^3+28 x^2+71 x+32$
• $y^2=18 x^6+35 x^5+26 x^4+26 x^3+44 x^2+12 x+43$
• $y^2=29 x^6+6 x^5+50 x^4+29 x^3+15 x^2+30 x+36$
• $y^2=54 x^6+56 x^5+10 x^4+26 x^3+40 x^2+59 x+16$
• $y^2=46 x^6+58 x^5+71 x^4+30 x^3+12 x^2+17 x+71$
• $y^2=x^6+58 x^5+66 x^4+38 x^3+47 x^2+12 x+63$
• $y^2=6 x^6+6 x^5+26 x^4+63 x^3+21 x^2+40 x+47$
• $y^2=16 x^5+59 x^4+71 x^3+53 x^2+54 x+2$
• $y^2=58 x^6+43 x^5+63 x^4+50 x^3+28 x^2+18 x+32$
• $y^2=35 x^6+41 x^5+23 x^4+19 x^3+55 x^2+18 x+68$
• $y^2=12 x^6+3 x^5+32 x^4+4 x^3+71 x^2+10 x+2$
• $y^2=40 x^6+59 x^5+65 x^4+70 x^3+54 x^2+12 x+5$
• $y^2=23 x^6+23 x^5+30 x^4+26 x^3+3 x^2+5 x+2$
• $y^2=69 x^6+42 x^5+62 x^4+56 x^3+33 x^2+25 x+64$
• $y^2=9 x^6+70 x^5+44 x^4+58 x^3+17 x^2+16 x+65$
• $y^2=29 x^6+72 x^5+28 x^4+30 x^3+64 x^2+50 x+27$
• $y^2=42 x^6+21 x^5+14 x^4+3 x^3+13 x^2+24 x+22$
• $y^2=30 x^6+69 x^5+7 x^4+13 x^3+52 x^2+7 x+3$
• $y^2=56 x^6+19 x^5+43 x^4+18 x^3+51 x^2+57 x+72$
• $y^2=4 x^6+24 x^5+48 x^4+43 x^3+41 x^2+31 x+68$
• $y^2=50 x^6+19 x^5+68 x^4+58 x^3+39 x^2+57 x+2$
• $y^2=46 x^6+19 x^5+42 x^4+26 x^3+51 x^2+70 x+23$
• $y^2=27 x^6+39 x^5+71 x^4+51 x^3+13 x^2+56 x+67$
• $y^2=4 x^6+31 x^5+x^4+45 x^3+5 x^2+47 x+70$
• $y^2=66 x^6+55 x^5+69 x^4+54 x^3+39 x^2+12 x+69$
• $y^2=35 x^6+26 x^5+50 x^4+41 x^3+45 x^2+58 x+61$
• $y^2=19 x^6+29 x^5+12 x^4+35 x^3+11 x^2+7 x+51$
• $y^2=45 x^6+60 x^5+33 x^4+15 x^3+28 x^2+67 x+67$
• $y^2=42 x^6+47 x^5+6 x^4+44 x^3+41 x^2+69 x+65$
• $y^2=33 x^6+50 x^5+41 x^4+4 x^3+30 x^2+39 x+62$
• $y^2=42 x^6+46 x^5+9 x^4+12 x^3+23 x^2+14 x+56$
• $y^2=67 x^6+7 x^5+18 x^4+64 x^3+37 x^2+x+26$
• $y^2=43 x^6+33 x^5+30 x^4+23 x^3+71 x^2+68 x+61$
• $y^2=10 x^6+56 x^5+37 x^4+8 x^3+26 x^2+18 x+54$
• $y^2=5 x^6+2 x^5+8 x^4+51 x^3+51 x^2+15 x$
• $y^2=37 x^6+30 x^5+70 x^4+18 x^3+39 x^2+22 x+72$
• $y^2=57 x^6+30 x^5+39 x^4+15 x^3+12 x^2+39 x+15$
• $y^2=68 x^6+66 x^5+72 x^4+47 x^3+47 x^2+71 x+67$
• $y^2=27 x^6+72 x^5+70 x^4+39 x^3+61 x^2+x+63$
• $y^2=9 x^6+54 x^5+16 x^4+68 x^3+15 x^2+2 x+40$
• $y^2=53 x^6+60 x^5+17 x^4+8 x^3+59 x^2+41 x+29$
• $y^2=64 x^6+58 x^5+5 x^4+70 x^3+36 x^2+62 x+48$
• $y^2=32 x^6+66 x^5+54 x^4+45 x^3+3 x^2+31 x+14$
• $y^2=3 x^6+23 x^5+56 x^4+12 x^3+25 x^2+30 x+6$
• $y^2=61 x^6+41 x^5+72 x^4+x^3+18 x^2+20 x+29$
• $y^2=67 x^6+4 x^5+16 x^4+71 x^3+71 x^2+58 x+37$
• $y^2=54 x^6+18 x^5+20 x^4+29 x^3+6 x^2+64 x+63$
• $y^2=13 x^5+67 x^4+48 x^3+72 x^2+x+45$
• $y^2=48 x^6+35 x^5+54 x^4+41 x^3+23 x^2+50 x+25$
• $y^2=25 x^6+17 x^5+23 x^4+60 x^3+44 x^2+38 x+29$
• $y^2=27 x^6+64 x^5+18 x^3+27 x^2+45 x+3$
• $y^2=55 x^6+13 x^5+69 x^4+25 x^3+35 x^2+12 x+67$
• $y^2=47 x^6+44 x^5+24 x^4+61 x^3+36 x^2+15 x+22$
• $y^2=65 x^6+56 x^5+72 x^4+69 x^2+61 x+6$

Decomposition and endomorphism algebra

All geometric endomorphisms are defined over $\F_{73}$.

Endomorphism algebra over $\F_{73}$

The isogeny class factors as 1.73.c $\times$ 1.73.n and its endomorphism algebra is a direct product of the endomorphism algebras for each isotypic factor. The endomorphism algebra for each factor is:

• 1.73.c : \(\Q(\sqrt{-2}) \).
• 1.73.n : \(\Q(\sqrt{-123}) \).

Base change

This is a primitive isogeny class.

Twists

Below is a list of all twists of this isogeny class.

Twist	Extension degree	Common base change
2.73.ap_gq	$2$	(not in LMFDB)
2.73.al_eq	$2$	(not in LMFDB)
2.73.l_eq	$2$	(not in LMFDB)